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TY - CHAP
TI - Identities for the Fibonacci Powers
A2 - Chen, Hongwei
AB - In this chapter, using generating functions, we establish the following identities for any positive integers*>n*and*k*,\[F_{n+k+1}^{k}=\sum\limits_{i=0}^{k}{{{a}_{i}}(k)F_{n+k-i,}^{k}}\]where the*a*_{i}(*k*) are given explicitly in terms of Fibonomial coefficients (see (12.7) below). Along the way, we will focus on how to derive identities, instead of merely focusing on verification.

Recall that Fibonacci numbers {*F*_{n}} are defined by\[\begin{align*} {{F}_{1}} & = {{F}_{2}}=1,\\ {{F}_{n+2}} & = {{F}_{n+1}}+{{F}_{n}}\quad (\text{for }n\ge 1).\end{align*}\caption {(12.1)}\]

In the previous chapter, we proved that for any*n*≥ 1,\[F_{n+3}^{2}=2F_{n+2}^{2}+2F_{n+1}^{2}-F_{n}^{2},\caption {(12.2)}\]\[F_{n+4}^{3}=3F_{n+3}^{3}+6F_{n+2}^{3}-3F_{n+1}^{3}-F_{n}^{3}.\caption {(12.3)}\]

In these identities, a power of a Fibonacci number is expressed as a linear combination of the same power of successive Fibonacci numbers. Naturally, we ask whether

EP - 142
ET - 1
PB - Mathematical Association of America
PY - 2010
SN - 9780883857687
SP - 131
T2 - Excursions in Classical Analysis
T3 - Pathways to Advanced Problem Solving and Undergraduate Research
UR - http://www.jstor.org/stable/10.4169/j.ctt5hh9r6.15
Y2 - 2021/03/04/
ER -